Finite Group Databasebeta

← #29593342

Subgroups — #29593342

Order 15616.

324 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456643C211yesmaximal, characteristic
2#21448038C212yesmaximal, complemented
3#21443812C212yesmaximal, complemented
4#21443755C211yesmaximal, characteristic
5C61×#2091C211yesmaximal, characteristic
6C2×#14519756—21no—
7#16840093—22no—
8#16840020—21no—
9#16840020C2212yes—
10#16839920—22no—
11#16839586—22no—
12#16839577—22no—
13#16839544C2212yes—
14C61×#430—21no—
15C61×#430C2211yescharacteristic
16C61×#407—21no—
17C61×#407C2211yescharacteristic
18C61×#338C2211yescharacteristic
19C22×Dic122—21no—
20C2×Dic244—21no—
21C2×Dic244D412yes—
22#14519756—22no—
23#14519756—22no—
24#14519756—41no—
25#14519756—41no—
26C2×C4⋊Dic61—21no—
27C2×#11758196—41no—
28C2×C22⋊Dic61—41no—
29#14519717—22no—
30#14519709—21no—
31#14519709D412yes—
32#14519702—22no—
33#14519702—22no—
34C23×C4×C61D411yescharacteristic
35C2×M16×C61—21no—
36C2×M16×C61—21no—
37C2×M16×C61D411yescharacteristic
38C2×M16×C61C2311yescharacteristic
39#108×C61—22no—
40#108×C61—42no—
41C22⋊C8×C61—22no—
42C22⋊C8×C61—42no—
43C24×C61SD1611yescharacteristic
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61—41no—
47C22×C4×C61—41no—
48C22×C4×C61SD1611yescharacteristic
49C22×C4×C61C2×D411yescharacteristic
50C22×Dic61—41no—
51C22×Dic61—41no—
52C2×Dic122—22no—
53C2×Dic122—41no—
54C2×Dic122—81no—
55C2×Dic122—81no—
56M16×C61—41no—
57M16×C61—41no—
58M16×C61—41no—
59M16×C61—41no—
60M16×C61—41no—
61M16×C61—41no—
62C2×C8×C61—22no—
63C2×C8×C61—21no—
64C2×C8×C61—41no—
65C2×C8×C61C2×D412yes—
66C22⋊Dic61—42no—
67C22⋊Dic61—42no—
68C4⋊Dic61—22no—
69C4⋊Dic61—41no—
70#11758196—42no—
71#11758196—42no—
72Dic244—42no—
73Dic244—81no—
74#11758185—42no—
75C23×C61—21no—
76C23×C61—41no—
77C23×C61—42no—
78C23×C61C2×SD1611yescharacteristic
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—41no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—81no—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
88C2×Dic61—42no—
89C2×Dic61—42no—
90C2×Dic61—82no—
91C2×Dic61—82no—
92Dic122—42no—
93Dic122—42no—
94Dic122—82no—
95C8×C61—22no—
96C8×C61—42no—
97C8×C61—41no—
98C8×C61—81no—
99#12299—611nomaximal, Sylow
100C22×C61—22no—
101C22×C61—21no—
102C22×C61—41no—
103C22×C61—41no—
104C22×C61—42no—
105C22×C61—41no—
106C22×C61—81no—
107C22×C61#45011yescharacteristic
108C4×C61—21no—
109C4×C61—41no—
110C4×C61#45311yescharacteristic
111C4×C61#45811yescharacteristic
112Dic61—82no—
113Dic61—82no—
114#3195—611no—
115#2239—612no—
116#2091D6111yescharacteristic
117#1342—612no—
118#1320—611no—
119C2×C61—21no—
120C2×C61—42no—
121C2×C61—41no—
122C2×C61#161311yescharacteristic
123C2×C22⋊Q8—1221no—
124#483—1222no—
125#470—612no—
126#470—1221no—
127#451—1222no—
128#430—21no—
129#430C2×D6111yescharacteristic
130#407—21no—
131#407C2×D6111yescharacteristic
132#365—1222no—
133#362—1222no—
134#349—612no—
135#338C2×D6111yescharacteristic
136C61#1229911yescharacteristic, complemented, Sylow
137C22×Q8—1221no—
138C23×C4C22⋊D6111yescharacteristic
139C2×Q16—612no—
140C2×Q16—1221no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16C22⋊D6111yescharacteristic
144C2×M16C22×D6111yescharacteristic
145C22⋊Q8—1222no—
146C22⋊Q8—1222no—
147C22⋊Q8—2441no—
148C22⋊Q8—2441no—
149C2×C4⋊C4—1221no—
150C2×C4⋊C4—2441no—
151C2×C22⋊C4—2441no—
152M32—1222no—
153#108—22no—
154#108—42no—
155#106—612no—
156#106—1221no—
157#103—1222no—
158#103—1222no—
159C22⋊C8—22no—
160C22⋊C8—42no—
161C24#1175820111yescharacteristic
162C2×Q8—1222no—
163C2×Q8—4881no—
164C2×Q8—4881no—
165C2×Q8—2441no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—2441no—
171C22×C4—2441no—
172C22×C4#1175820111yescharacteristic
173C22×C4C2×C22⋊D6111yescharacteristic
174Q16—4881no—
175Q16—2442no—
176M16—41no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—41no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8D4×D6112yes—
186C4⋊C4—1222no—
187C4⋊C4—2441no—
188C4⋊C4—2442no—
189C4⋊C4—2442no—
190C22⋊C4—2442no—
191C22⋊C4—2442no—
192C16—2442no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1175820111yescharacteristic
197Q8—4882no—
198Q8—2442no—
199Q8—2442no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—4882no—
208C2×C4—4882no—
209C2×C4—2442no—
210C2×C4—2442no—
211C2×C4#1451974511yescharacteristic
212C2×C4#1451980211yescharacteristic
213C8—22no—
214C8—42no—
215C8—41no—
216C8—81no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1683991211yescharacteristic
225C4—21no—
226C4—41no—
227C4—4882no—
228C4—4882no—
229C4#1683992911yescharacteristic
230C4#1683995111yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2144483811yescharacteristic
subgroupG/H
centerC2#21444838
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#2091C2
socleC2×C61#1613
prime radicalC24×C61SD16
hypercenter#2091D61
hypocenterC61#12299
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2091

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21448038C2semidirect
3. #21443812C2semidirect
136. C61#12299semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214566432
#214480382
#214438122
#214437552
C61×#20912
#1229961